Integrand size = 29, antiderivative size = 101 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\frac {17 a^4 x}{2}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {32 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {a^4 \cos (c+d x) \sin (c+d x)}{2 d} \]
17/2*a^4*x-4*a^4*cos(d*x+c)/d+4/3*a^4*cos(d*x+c)/d/(1-sin(d*x+c))^2-32/3*a ^4*cos(d*x+c)/d/(1-sin(d*x+c))-1/2*a^4*cos(d*x+c)*sin(d*x+c)/d
Time = 7.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.56 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=-\frac {a^4 \left (-3 (161+204 c+204 d x) \cos \left (\frac {1}{2} (c+d x)\right )+(647+204 c+204 d x) \cos \left (\frac {3}{2} (c+d x)\right )-39 \cos \left (\frac {5}{2} (c+d x)\right )+3 \cos \left (\frac {7}{2} (c+d x)\right )+6 (146+136 c+136 d x+(-59+68 c+68 d x) \cos (c+d x)-14 \cos (2 (c+d x))-\cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
-1/48*(a^4*(-3*(161 + 204*c + 204*d*x)*Cos[(c + d*x)/2] + (647 + 204*c + 2 04*d*x)*Cos[(3*(c + d*x))/2] - 39*Cos[(5*(c + d*x))/2] + 3*Cos[(7*(c + d*x ))/2] + 6*(146 + 136*c + 136*d*x + (-59 + 68*c + 68*d*x)*Cos[c + d*x] - 14 *Cos[2*(c + d*x)] - Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(d*(Cos[(c + d*x) /2] - Sin[(c + d*x)/2])^3)
Time = 0.34 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3351, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \tan ^2(c+d x) \sec ^2(c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 (a \sin (c+d x)+a)^4}{\cos (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle a^4 \int \left (\sin ^2(c+d x)+4 \sin (c+d x)-\frac {12}{1-\sin (c+d x)}+\frac {4}{(1-\sin (c+d x))^2}+8\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 \left (-\frac {4 \cos (c+d x)}{d}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}-\frac {32 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {17 x}{2}\right )\) |
a^4*((17*x)/2 - (4*Cos[c + d*x])/d + (4*Cos[c + d*x])/(3*d*(1 - Sin[c + d* x])^2) - (32*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (Cos[c + d*x]*Sin[c + d*x])/(2*d))
3.9.20.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.31
| method | result | size |
| risch | \(\frac {17 a^{4} x}{2}+\frac {i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {2 a^{4} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {2 a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{d}-\frac {i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {8 \left (-15 i a^{4} {\mathrm e}^{i \left (d x +c \right )}+9 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-8 a^{4}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}\) | \(132\) |
| parallelrisch | \(-\frac {a^{4} \left (-204 d x \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-204 d x \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-612 d x \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+612 d x \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+3 \sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+39 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+575 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+39 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-3 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )-207 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+267 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-837 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \left (\sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )\right )}\) | \(193\) |
| derivativedivides | \(\frac {a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+4 a^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+6 a^{4} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+4 a^{4} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(268\) |
| default | \(\frac {a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+4 a^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+6 a^{4} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+4 a^{4} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(268\) |
| norman | \(\frac {-\frac {17 a^{4} x}{2}+\frac {80 a^{4}}{3 d}+\frac {17 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {26 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {307 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {188 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {307 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {26 a^{4} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {17 a^{4} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {17 a^{4} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {51 a^{4} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {51 a^{4} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {51 a^{4} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {51 a^{4} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {17 a^{4} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {17 a^{4} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {16 a^{4} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {96 a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {80 a^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {304 a^{4} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {544 a^{4} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(392\) |
17/2*a^4*x+1/8*I*a^4/d*exp(2*I*(d*x+c))-2*a^4/d*exp(I*(d*x+c))-2*a^4/d*exp (-I*(d*x+c))-1/8*I*a^4/d*exp(-2*I*(d*x+c))-8/3*(-15*I*a^4*exp(I*(d*x+c))+9 *a^4*exp(2*I*(d*x+c))-8*a^4)/(exp(I*(d*x+c))-I)^3/d
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (89) = 178\).
Time = 0.26 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.95 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\frac {3 \, a^{4} \cos \left (d x + c\right )^{4} - 18 \, a^{4} \cos \left (d x + c\right )^{3} - 102 \, a^{4} d x - 8 \, a^{4} + 17 \, {\left (3 \, a^{4} d x + 5 \, a^{4}\right )} \cos \left (d x + c\right )^{2} - {\left (51 \, a^{4} d x - 98 \, a^{4}\right )} \cos \left (d x + c\right ) - {\left (3 \, a^{4} \cos \left (d x + c\right )^{3} - 102 \, a^{4} d x + 21 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - {\left (51 \, a^{4} d x - 106 \, a^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
1/6*(3*a^4*cos(d*x + c)^4 - 18*a^4*cos(d*x + c)^3 - 102*a^4*d*x - 8*a^4 + 17*(3*a^4*d*x + 5*a^4)*cos(d*x + c)^2 - (51*a^4*d*x - 98*a^4)*cos(d*x + c) - (3*a^4*cos(d*x + c)^3 - 102*a^4*d*x + 21*a^4*cos(d*x + c)^2 + 8*a^4 - ( 51*a^4*d*x - 106*a^4)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^2 - d*co s(d*x + c) + (d*cos(d*x + c) + 2*d)*sin(d*x + c) - 2*d)
Timed out. \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.56 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\frac {2 \, a^{4} \tan \left (d x + c\right )^{3} + {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{4} + 12 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4} - 8 \, a^{4} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {8 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{4}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \]
1/6*(2*a^4*tan(d*x + c)^3 + (2*tan(d*x + c)^3 + 15*d*x + 15*c - 3*tan(d*x + c)/(tan(d*x + c)^2 + 1) - 12*tan(d*x + c))*a^4 + 12*(tan(d*x + c)^3 + 3* d*x + 3*c - 3*tan(d*x + c))*a^4 - 8*a^4*((6*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 + 3*cos(d*x + c)) - 8*(3*cos(d*x + c)^2 - 1)*a^4/cos(d*x + c)^3)/d
Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.34 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\frac {51 \, {\left (d x + c\right )} a^{4} + \frac {6 \, {\left (a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {16 \, {\left (6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \]
1/6*(51*(d*x + c)*a^4 + 6*(a^4*tan(1/2*d*x + 1/2*c)^3 - 8*a^4*tan(1/2*d*x + 1/2*c)^2 - a^4*tan(1/2*d*x + 1/2*c) - 8*a^4)/(tan(1/2*d*x + 1/2*c)^2 + 1 )^2 + 16*(6*a^4*tan(1/2*d*x + 1/2*c)^2 - 15*a^4*tan(1/2*d*x + 1/2*c) + 7*a ^4)/(tan(1/2*d*x + 1/2*c) - 1)^3)/d
Time = 17.13 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.84 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^4 \tan ^2(c+d x) \, dx=\frac {17\,a^4\,x}{2}+\frac {\frac {17\,a^4\,\left (c+d\,x\right )}{2}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {51\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (153\,c+153\,d\,x-378\right )}{6}\right )-\frac {a^4\,\left (51\,c+51\,d\,x-160\right )}{6}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {51\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (153\,c+153\,d\,x-102\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {85\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (255\,c+255\,d\,x-306\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {85\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (255\,c+255\,d\,x-494\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {119\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (357\,c+357\,d\,x-460\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {119\,a^4\,\left (c+d\,x\right )}{2}-\frac {a^4\,\left (357\,c+357\,d\,x-660\right )}{6}\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
(17*a^4*x)/2 + ((17*a^4*(c + d*x))/2 - tan(c/2 + (d*x)/2)*((51*a^4*(c + d* x))/2 - (a^4*(153*c + 153*d*x - 378))/6) - (a^4*(51*c + 51*d*x - 160))/6 + tan(c/2 + (d*x)/2)^6*((51*a^4*(c + d*x))/2 - (a^4*(153*c + 153*d*x - 102) )/6) - tan(c/2 + (d*x)/2)^5*((85*a^4*(c + d*x))/2 - (a^4*(255*c + 255*d*x - 306))/6) + tan(c/2 + (d*x)/2)^2*((85*a^4*(c + d*x))/2 - (a^4*(255*c + 25 5*d*x - 494))/6) + tan(c/2 + (d*x)/2)^4*((119*a^4*(c + d*x))/2 - (a^4*(357 *c + 357*d*x - 460))/6) - tan(c/2 + (d*x)/2)^3*((119*a^4*(c + d*x))/2 - (a ^4*(357*c + 357*d*x - 660))/6))/(d*(tan(c/2 + (d*x)/2) - 1)^3*(tan(c/2 + ( d*x)/2)^2 + 1)^2)